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Fuzzy hidden Markov chains segmentation for volume

determination and quantitation in PET.

Mathieu Hatt, Frederic Lamare, Nicolas Boussion, Christian Roux, Alexandre

Turzo, Catherine Cheze-Lerest, P. Jarritt, K. Carson, Fabien Salzenstein,

Christophe Collet, et al.

To cite this version:

Mathieu Hatt, Frederic Lamare, Nicolas Boussion, Christian Roux, Alexandre Turzo, et al..Fuzzy hidden Markov chains segmentation for volume determination and quantitation in PET..Physics in Medicine and Biology, IOP Publishing, 2007, 52 (12), pp.3467-91. <10.1088/0031-9155/52/12/010>. <inserm-00150348>

HAL Id: inserm-00150348

http://www.hal.inserm.fr/inserm-00150348

Submitted on 6 Apr 2009

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1

Fuzzy hidden Markov chains segmentation for volume

determination and quantitation in PET

M. Hatt1, F. Lamare1, N. Boussion1, A. Turzo1,2, C. Collet3, F. Salzenstein4,

C. Roux5,1, P. Jarritt6, K. Carson6, C. Cheze-Le Rest1,2, D. Visvikis1.

1 INSERM, U650, LaTIM, Brest, F-29200 France.

2 Academic Department of Nuclear Medicine, CHU Morvan, Brest, F-29609 France

3 Ecole Nationale Supérieure de Physique de Strasbourg (ENSPS), ULP, Strasbourg, F-67000 France.

4 Institut d'Électronique du Solide et des Systèmes (InESS), ULP, Strasbourg, F-67000 France.

5 ENST Bretagne, GET-ENST, Brest, F-29200 France

6 Medical Physics Agency, Royal Victoria Hospital, Belfast, Northern Ireland.

Corresponding author:

Mathieu HATT, INSERM U650, Laboratoire du Traitement de l’Information Médicale (LaTIM) CHU MORVAN, Bat 2bis (I3S), 5 avenue Foch, Brest, 29609 France Tel: +33 2 98 01 81 11 Fax: +33 2 98 01 81 24

2

ABSTRACT

Accurate volume of interest (VOI) estimation in PET is crucial in different oncology

applications such as response to therapy evaluation and radiotherapy treatment planning. The objective

of our study was to evaluate the performance of the proposed algorithm for automatic lesion volume

delineation; namely the Fuzzy Hidden Markov Chains (FHMC), with that of current state of the art in

clinical practice threshold based techniques. As the classical Hidden Markov Chain (HMC) algorithm,

FHMC takes into account noise, voxel’s intensity and spatial correlation, in order to classify a voxel as

background or functional VOI. However the novelty of the fuzzy model consists of the inclusion of an

estimation of imprecision, which should subsequently lead to a better modelling of the “fuzzy” nature

of the object on interest boundaries in emission tomography data. The performance of the algorithms

has been assessed on both simulated and acquired datasets of the IEC phantom, covering a large range

of spherical lesion sizes (from 10 to 37mm), contrast ratios (4:1 and 8:1) and image noise levels. Both

lesion activity recovery and VOI determination tasks were assessed in reconstructed images using two

different voxel sizes (8mm3 and 64mm3). In order to account for both the functional volume location

and its size, the concept of % classification errors was introduced in the evaluation of volume

segmentation using the simulated datasets. Results reveal that FHMC performs substantially better

than the threshold based methodology for functional volume determination or activity concentration

recovery considering a contrast ratio of 4:1 and lesion sizes of <28mm. Furthermore differences

between classification and volume estimation errors evaluated were smaller for the segmented

volumes provided by the FHMC algorithm. Finally, the performance of the automatic algorithms was

less susceptible to image noise levels in comparison to the threshold based techniques. The analysis of

both simulated and acquired datasets led to similar results and conclusions as far as the performance of

segmentation algorithms under evaluation is concerned.

3

1. Introduction

Positron Emission Tomography (PET) has been long established as a powerful tool in

oncology, particularly in the area of diagnosis. However, alternative applications such as the use of

PET in radiotherapy planning (Jarritt et al 2006) and response to therapy studies (Krak et al 2005) are

rapidly gaining ground. Whereas accurate activity concentration recovery is crucial for correct

diagnosis and monitoring response to therapy, applications such as the use of PET in Intensity-

Modulated Radiation Therapy (IMRT) treatment planning renders equally vital the accurate shape and

volume determination of lesions. Different volume of interest (VOI) determination methodologies

have been proposed that can be classified as manual or automatic. On the one hand, manual

segmentation methods to delineate boundaries are laborious and highly subjective (Krak et al 2005).

On the other hand, automatic segmentation of objects of interest in PET (Reutter et al 1997, Zhu et al

2003, Kim et al 2003, Riddell et al 1999) is not a trivial task because of low spatial resolution and

resulting partial volume effects (PVE), low contrast ratios, as well as noise resulting from the

statistical nature of radioactive decay or the choice of the reconstruction process.

The most widely used method to semi-automatically determine VOIs in PET is thresholding,

either adaptive, using a priori Computed Tomography (CT) knowledge (Erdi et al 1997), or fixed

threshold (Krak et al 2005) using values derived from phantom studies (from 30 to 75% of maximum

local activity concentration value) (Jarritt et al 2006, Krak et al 2005, Erdi et al 1997). Such

thresholding techniques however lead to variable VOIs determination as shown in recent clinical

studies (Nestle et al 2005). On the other hand, numerous works have addressed automatic lesion

detection from PET datasets, including different methodologies such as edge detection (Reutter et al

1997), fuzzy C-Means (Zhu et al 2003), clustering (Kim et al 2003) or watersheds (Riddell et al 1999).

The performance of these algorithms is sensitive to variations of noise intensity and/or lesion contrast.

In addition, past work has in its majority considered the ability of such automatic methodologies for

the detection of lesions but not the accuracy with which the methods are capable for VOI and/or

activity concentration determination. Furthermore, all of the afore-mentioned algorithms often involve

user-dependent initializations, pre- and post-processing, or additional information like CT or expert

4

knowledge; rendering their employment more complicated and the outcome dependent on choices

made by the user in relation to the pre- and/or post-processing steps necessary. For example in the case

of the watershed algorithm a filtering pass as pre-processing step to smooth the image, and a post-

processing step to fuse the different regions resulting from the algorithm are necessary.

Hidden Markov Models are automatic segmentation algorithms allowing noise modelling and

have proven to be less sensitive to variation of the values in the regions of the images than other

segmentation approaches thanks to their statistical modelling. They have only been previously used in

PET in the form of Hidden Markov Fields (HMF) (Chen et al 2001). Hidden Markov Chains (HMC)

(Benmiloud et al 1995) is a faster model and can offer competitive results (Salzenstein et al 1998).

Furthermore, HMC leads to shorter computational times, as quantities of interest can be computed

directly on the chain, whereas the HMF algorithm needs iterative Monte-Carlo like estimation

procedures (Salzenstein et al 1998) that are time consuming. These algorithms offer an unsupervised

estimation of the parameters needed for the image segmentation and limit the user’s input to the

number of classes to be searched for in the image. Reconstructed images require no further pre- or

post-processing treatment (such as for example filtering) prior to the segmentation process. Instead,

image noise is considered as additional information (a parameter in the classification decision process)

to be taken into account, not to be suppressed or avoided.

The objectives of our study were to (a). develop a new fuzzy HMC (FHMC) model in an

attempt to account for the limited spatial resolution in PET and (b) compare the performance of

FHMC with these of the thresholding methodologies currently used in clinical practice. Different

imaging conditions in terms of statistical quality, as well as lesion size and source-to-background

(S/B) ratio were considered in this study. The analysis was carried out on both simulated and acquired

images reconstructed using iterative algorithms which form today’s state of the art in whole body PET

imaging in routine clinical oncology practice (Visvikis et al 2001, Visvikis et al 2004).

2. Materials and methods

2.1 Hard and Fuzzy Hidden Markov Chain models

5

The HMC model is an unsupervised methodology that takes place in the Bayesian framework.

Although we place ourselves in the application of image segmentation this methodology can be used

in other applications such as for example speech recognition (Dai 1994). Let T be a finite set

corresponding to the voxels of an image. We consider two random processes Y ( )t t Ty ∈= and

X ( )t t Tx ∈= . Y represents the observed image, and X represents the “hidden” segmentation map. X

takes its values in 1,..., KΩ = with K being the number of classes c , and Y takes its values in .

We assume that a Markov process can model the prior distribution of X . The segmentation problem

consists in estimating the hidden X from the available noisy observation Y . The relationship

between X and Y can be modelled by the joint distribution (X,Y)P . This distribution can be

obtained thanks to the Bayes formula:

(X,Y) (Y | X) (X)

(X | Y)(Y) (Y)

P P PP

P P= = (1).

(Y|X)P is the likelihood of the observation Y conditionally with respect to the hidden ground-truth

X , and (X)P is the prior knowledge concerning X . The Bayes rule allows us to know the posterior

distribution of X with respect to the observation Y . In the Markov Chain framework we have to

assume the random variables Y ( )t t Ty ∈= are conditionally independent with respect to X and that

the distribution of each ty conditional on X is equal to its distribution conditional on tx . Many

applications of Hidden Markov Models with unsupervised estimation have been successful

considering different types of images (radar, sonar, Magnetic Resonance Images (MRI), CT, satellite

or astronomical) (Pieczynski 2003, Salzenstein et al 2004, Delignon 1997), but this kind of approach

was almost never applied on PET data.

2.1.1 Markov Chain definition

X is a Markov chain if:

1 1 1( | , ..., ) ( | ) for 1t t t t

P x x x P x x t T− −= < ≤ (2)

The distribution of X is then defined by the distribution of 1x , called initial probabilities ( )init c for

each class c ( 1( )P x c= ) and the transition matrix ( , )trans c d (of dimension K K× ) containing the

6

probabilities of transitions from the class c to the class d ; 1( | )t tP x d x c+ = = . As X and Y are

one-dimensional elements in the HMC context, a spatial transformation is necessary to process three-

dimensional VOIs. For the best preservation of the spatial correlation between voxels we use the

Hilbert-Peano space-filling curve. This fractal path can be extended to explore 3D VOIs (Kamata et al

1999). A visual illustration of the Hilbert-Peano path for a 4× 4×4 voxels 3D VOI is given in figure 1.

Once the chain has been segmented, the inverse path is used to reconstruct the 3D segmentation map.

2.1.2 Adding a fuzzy measure to the model

The general idea behind the implementation of a fuzzy model within the Bayesian framework

was previously introduced by Salzenstein (Salzenstein et al 1997). Its implementation in association

with HMC developed as part of this work is based on the incorporation of a finite number of fuzzy

levels iF in combination with two homogeneous (or “hard”) classes, in comparison to HMC where

only a finite number of hard classes are considered. This model allows the coexistence of voxels

belonging to one of two hard classes and voxels belonging to a “fuzzy level” depending on its

membership to the two hard classes. Therefore, FHMC adds an estimation of imprecision of the

hidden data (X, see section 2.1) in contrast to HMC which only models uncertainty of the observed

data (Y, see section 2.1). The statistical part of the algorithm models the uncertainty of the

classification, with the assumption being that the voxel is clearly identified but the observed data is

noisy. On the other hand, the fuzzy part models the imprecision of the voxel’s membership, with the

assumption being that the voxel may contain both classes. One way to achieve this extension is to

simultaneously use Dirac and Lesbegue measures at the class chain level. Hence we consider that X

in the fuzzy model takes its values in [ ]0,1 instead of 1,..., KΩ = . Let 0δ and 1δ be the Dirac

measures at 0 and 1, and ζ the Lesbegue measure on ] [0,1 . We define the new measure 0 1ν δ δ ζ= + +

on [ ]0,1 . Note that, for example, using two hard classes and two fuzzy levels in the FHMC model is

not equivalent to using four hard classes in the HMC model where 1 2 ... Kν δ δ δ= + + + . This has been

previously stated using Markov Fields based segmentation (Salzenstein et al 1997).

7

The distribution of X can then be defined using a conjoint density g for 1( , )t tx x + on [ ] [ ]0,1 0,1× :

let [ ] [ ]( , ) 0,1 0,1a b ∈ ×

1 2

1 2

( 0, 0) and ( 1, 1)

( 0, 1) and ( 1, 0)

g a b g a b

g a b g a b

α α

γ γ

= = = = = =

= = = = = =

( , ) ( , ) if ( , ) (0,0),(0,1),(1,0),(1,1)gg a b f a b a bβ= ≠ (3)

With [ ][ ]0,1 0,1

( , ) ( )( , ) 1g a b d a bν ν⊗ =∫ ∫ and 1 2 1 2 1α α γ γ βλ+ + + + = (4)

where, ( )( , )d a bν ν⊗ is the notation for integration with respect to the (a,b) variables, each one

being with respect to the measure ν on the interval [0,1]. λ is a constant depending on the form of the parameterized function gf :

( , ) 1- | - |gf a b a b= (5)

We now define the initial and transition probabilities ( ( ) and ( , )init c trans c d ) using the conjoint

density g and an utility density h on [ ]0,1 defined by: 1

0

( ) ( , ) ( )h a g a b d bν= ∫ :

( )init c using densities g and h :

( )

( )

1

1 1 1

0

1 1

11 0 0

0,1 ( , ) ( ) ( )

1 1( , ) ( )( , ) ( , ) ( )( ) ( )

i

N

i i i

i

N

P x g x b d b h x

P x F g a b d a b g b d b hN N

ν

ν ν ε ν ε−

∈ = =

∈ = ⊗ =

∫

∫ ∫ ∫

(6)

( , )trans c d using the conditional density f deduced from (1) : 11

( , ) ( | )

( )t t

t t

t

g x xf x x

h x

++ =

( )

( )

( )

( )

1

1

1

1

11

( , ) ( )( , )|

( ) ( )

( , ) ( )| 0,1

( )

( , ) ( )0,1 |

( ) ( )

( , )0,1 | 0,1

( )

j i

i

j

i

i

j i j iF F

t j t i

i iF

j t jF

t j t

t

t i iF

t t i

i iF

t tt t

t

g d

P x F x Fh d

g x d

P x F xh x

g x dP x x F

h d

g x xP x x

h x

ε ε ν ν ε ε

ε ν ε

ε ν ε

ε ν ε

ε ν ε

+

+

+

+

++

⊗∈ ∈ =

∈ ∈ =

∈ ∈ =

∈ ∈ =

∫ ∫

∫

∫

∫

∫

(7)

where 1N − is the number of fuzzy levels and i

i

Nε = is the value associated to a fuzzy level iF .

The fuzzy model is a generalization of the hard model. The use of the Dirac measures allows

one to retrieve the standard hard model when the fuzzy component is null. As the theoretical

framework described above has not been developed for a specific kind of image, but as a general

8

segmentation algorithm, the a priori and the noise (also called observation) models are not directly

derived from PET image characteristics. However this segmentation approach may be appropriate in

segmenting PET images since they are both noisy and of low resolution. The “noise” aspect when

considering Hidden Markov Models in general is the way the values of each class to be found in the

image are distributed around a mean value. The noise model used, whose respective mean and

variance are to be determined by the estimation steps, can therefore be adapted to image specific

characteristics. On the other hand, the fuzzy measure allows a more realistic modelling of the objects’

borders transitions between foreground and background, allowing in such a way to indirectly account

for the effects of blurring (partial volume effects) associated with low resolution images, such as those

in PET.

2.1.3 Segmentation and parameters estimation

In order to perform segmentation on the chain level, we need to use a criterion to classify each

element as background or functional VOI. For this purpose we use the Marginal Posterior Mode

(MPM) (Marroquin et al 1987). This approach aims to minimize the expectation ( , ) |t tE L x x Y

where L is a loss (or cost) function:

ˆ ˆ( , )t t t tL x x x x= − (8)

With tx the real class and ˆ ˆ( )t tx s y= the one affected by the segmentation process s . This criterion

is adequate for the segmentation problem as it penalizes a configuration with respect to the number of

misclassified elements. In order to compute a solution, the MPM segmentation needs the parameters

defining the a priori model (initial and transition probabilities of the chain) as well as the noisy

observation data model (mean and variance of each class). The assumption that the noise for each class

of the observed data can fit a Gaussian distribution was made as a first step. The mean and variance of

each fuzzy level iF is derived from the ones estimated in the two hard classes as follows:

0 1

2 2 2 2 20 1

(1 )

(1 )

i

i

F i i

F i i

µ µ ε ε µ

σ σ ε ε σ

= − +

= − + (9)

9

Both a priori and noise models parameters are unknown in the real case and therefore they

must be estimated. In order to achieve such estimation, we use the stochastic iterative procedure called

Stochastic Expectation Maximization (SEM) (Celeux & Diebolt 1986), a stochastic version of the EM

algorithm (Dempster 1977). This is achieved in a similar fashion to that used in the classical HMC

context by simulating posterior realizations of X (see Appendix for detailed posterior realization of X

and the SEM procedure) and computing empirical values of the parameters of interest using the

simulated chain. The stochastic nature of this procedure makes it less sensitive to the initial guess of

the parameters using the K-Means (McQueen 1967) than deterministic procedures like the EM

algorithm. Both the MPM segmentation and SEM parameters estimation use a practical recursive

computation of the values of interest called Forward-Backward procedure that is performed directly on

the chain (Benmiloud et al 1995). The implementation of the FHMC segmentation algorithm in a step

by step fashion can be found in the Appendix. Note that the overall algorithm is entirely unsupervised

(except for the number of classes and fuzzy levels to use) and it is able to adjust to a large spectrum of

image structures, noise or contrast. For example, no a priori is made on the shape of the objects to

extract or the source-to-background ratio in the image.

2.2 Thresholding

Various thresholding methodologies have been proposed in the past for both functional

volume segmentation and/or activity concentration recovery (Krak et al 2005, Erdi et al 1997, Nestle

et al 2005). Thresholding using 42% and 50% of the maximum value in the lesion was chosen for VOI

determination and quantitation purposes respectively, based on previous publications (Krak et al 2005,

Erdi et al 1997). The methodology was implemented through region growing using the voxel of

maximum intensity in the object of interest as a seed. Using a 3-D neighbourhood (26 neighbours) the

region is iteratively increased by adding neighbouring voxels if their intensity is superior or equal to

the selected threshold value. The results derived using these methods will be denoted from here

onwards as T42 and T50 for the thresholds of 42% and 50% respectively.

2.3 Validation studies

10

2.3.1 Simulated and acquired datasets

Simulated datasets using the IEC image quality phantom (Jordan 1990), containing six

different spherical lesions of 10, 13, 17, 22, 28 and 37 mm in diameter (figure 2), were generated

using Geant4 Application for Tomographic Emission (GATE) and a validated model of the Philips

Allegro PET scanner (Lamare et al 2006). Images, considering only the detected true coincidences,

were subsequently reconstructed using the OPL-EM iterative algorithm (Reader et al 2002) with 7

iterations (Lamare et al 2006). Two different voxel sizes were considered in the reconstructed images;

namely 2×2×2 mm3 and 4×4× 4 mm3. The 8 mm3 voxel size configuration leads to better sampled

objects of interest but with higher noise due to the number of counts being divided by eight in each

voxel in comparison to the 64 mm3 voxel sizes. A uniform activity was simulated throughout the

phantom cylinder and the lesions. Different parameters were however considered to cover a large

spectrum of configurations allowing assessment of the influence of different parameters susceptible to

affect the functional VOI determination or quantitation accuracy. The statistical quality of the images

was varied by considering 20, 40 and 60 millions of true coincidences. Two different signal to

background (S/B) ratios were also considered; 4:1 and 8:1 (with around 6kBq/cm3 in the background,

and 24 or 48kBq/cm3 in the spheres respectively). Visual illustration of the reconstructed images

corresponding to different simulated configurations is given in figure 3(a)-(d).

In addition to the simulated datasets, acquisitions of the IEC phantom were carried out in list

mode format using a Philips GEMINI PET/CT scanner. The only difference with the simulated

datasets was the exclusion of the 28 mm diameter sphere in the study because in the phantom used it

was replaced by a plastic sphere of unknown diameter. The same S/B ratios of 4:1 and 8:1 used in the

simulations were also employed in this part of the study, by introducing 7.4kBq/cm3 in the background

and 29.6 or 59.2kBq/cm3 respectively in the spheres. Different count statistical qualities were obtained

by reconstructing 1 min, 2min or 5min list-mode time frames using the 3D RAMLA algorithm, with

specific parameters previously optimised (Visvikis et al 2004). The same voxel sizes as for the

simulated datasets (8 mm3 and 64 mm3) were used in the reconstruction of each of the different

statistical quality datasets considered. Visual illustration of the acquired images is given in figure 3(e)-

11

(h). Each sphere in both simulated and acquired images was isolated in a box of the same size

(16× 16× 10 for the 4mm case, and 32× 32× 20 for the 2mm case) prior to the segmentation process.

2.3.2 Computed volume versus classification error measurement

The majority of previous works dealing with VOI determination in PET measure the

performance of a given methodology by computing the VOI obtained on the segmentation map and

comparing it with the true known volume of the object of interest. This type of approach has the

potential of leading to biased performance measurements since a segmentation result may contain two

different types of error. On the one hand, one may have voxels of the background that are classified as

belonging to the object of interest, denoted from here on as positive classification errors (PCE), while

on the other hand, one may end up with voxels of the object that are classified as belonging to the

background, denoted from here on as negative classification errors (NCE). These classification errors

essentially occur on the boundaries of the objects of interest because of “spill in” (increasing

probabilities of a NCE) and “spill out” (increasing probabilities of a PCE). If the segmentation results

in PCEs and NCEs of equal amounts, the computed VOI would be very close to the true known

volume whereas the shape and position of the object would be incorrect. The shape and position

information is as important as the total volume of the object in order to accurately derive a

radiotherapy treatment planning or the activity concentration of interest in a response to therapy study

based on the derived functional volume. For example, let us assume that the segmentation process

results in 20% NCEs and 15% PCEs. This leads to a classification error of 35% whereas the error in

the overall computed volume is only -5%. Hence, the use of classification error is a more pertinent

measurement of the accuracy with which a given algorithm performs the task of functional volume

delineation since it takes into account not only the segmented volume in comparison to the actual

volume of interest but also its position and shape.

In the simulation study the total number of PCEs and NCEs is considered with respect to the

number of voxels defining the sphere (VoS) in the digital phantom (the ground truth) in order to obtain

a percentage classification error (CE):

12

( )

100PCE NCE

CEVoS

+= × (10)

The size of classification errors can be bigger than 100% in the case where a large number of

background voxels in the selected area of interest are misclassified as belonging to the sphere. In

practical terms, maximum classification errors calculated during this work where limited to 200%,

since any such values represent complete failure of the segmentation process. In addition, the interest

of classification errors is when they occur at the borders of the objects and not in other regions of the

background. One should also keep in mind that a combined representation of PCE and NCE into CE

leads to a loss of information as far as the direction of the bias is concerned. It does however still

represent more pertinent information than overall volume estimation errors, which reflect neither

accurate magnitude nor direction of the bias for a segmented volume.

On the other hand in the case of the images reconstructed from the acquired datasets only

overall computed volumes were considered in order to avoid any biases as a result of misalignment

and rescaling inaccuracies, as well as reconstruction artefacts in the higher and lower slices of the

associated CT datasets. As the goal is not to detect the lesion in the whole image but to estimate its

volume, shape and position with the best accuracy possible, we assume that the lesion has been

previously identified by the clinician and automatically or manually placed in a 3-D “box” well

encompassing the object. Subsequently, the images of the selected area were segmented in two classes

(functional VOI and background) using each of the three methods under evaluation (thresholding,

FHMC and HMC). In the FHMC case, different numbers of fuzzy levels were considered in the

segmentation process (namely 2 and 3). Following the segmentation by FHMC, volumes of interest

can be defined using the hard classes and any number of the fuzzy levels considered.

2.3.3 Quantitation Accuracy

In terms of quantitation the objective of our study was to determine the accuracy of the

average activity concentration recovered from a volume derived using a given segmentation algorithm.

The “ground truth” for comparison purposes was established using the exact size, shape and location

of each lesion (using the known digital phantom employed in the generation of the simulated datasets).

13

As a result, these recovered activity concentration values represented an under-estimation of

the true activity due to PVE. A comparison on a lesion by lesion basis was subsequently carried out

with the measured activity concentration from the segmented volumes obtained by the three

algorithms considered. T50 should lead to some improvements in the lesion activity recovery with

respect to T42 as a result of including less voxels in the volume used to compute the activity and

therefore less voxels associated with PVE. Similarly FHMC 0/2 (see section 3. Results for the

definition of FHMC x/y) should lead to concentration recovery improvements with respect to FHMC

1/2, since voxels belonging to the fuzzy levels are found at the edges of the lesions and their intensity

is most significantly reduced by PVE. Therefore the inclusion of these voxels should only result in

even stronger under-evaluation of the true lesion activity concentrations.

3. Results

Different segmentation maps obtained using each of the methods under evaluation are

presented in figure 4 for a slice centred on the 28 mm sphere of the simulated images to visually

illustrate the variations of the segmentation maps obtained. Figure 5(a) shows the impact of the

number of fuzzy levels included in the FHMC segmentation. The various FHMC maps are denoted as

FHMC x/y with x being the number of fuzzy levels included in the segmentation map, and y being the

total number of fuzzy levels used in the segmentation process. The error bars in these figures represent

different results obtained for each of the 3 different levels of statistical quality considered (the top of

the error bar is the result concerning the worst statistical quality, the medium one concerns the

medium quality, and the lowest one corresponds to the best quality considered). As Figure 5(a) shows,

for the range of simulated spheres considered, no improvement was obtained in the % classification

errors by having more than 2 fuzzy levels in the FHMC segmentation process and keeping in the

overall segmented volume more than the voxels identified in the first fuzzy level. It should be

emphasized at this point, that this conclusion was reached considering the results on the whole of the

range of simulated sphere diameters and keeping in mind that our objective is determining a single

best configuration of the algorithm parameters across a wide range of imaging conditions and not

different parameters for individual lesion sizes, image statistics or contrast ratios. In addition, it is

14

clearly showed in figure 5 that HMC leads to worse segmentation results in comparison to FHMC for

all different configurations considered. Therefore for all subsequent volume determination analyses,

the results associated with the FHMC 1/2 versus T42 are presented. As shown in figure 5(b), no

benefits are observed through the inclusion in the segmentation map of any voxels belonging to the

fuzzy domain. This confirms what was anticipated in section 2.3.3. Therefore from here onwards all

the quantitation results presented for FHMC have been calculated using only the hard class voxels

resulting from the segmentation process (FHMC 0/2).

The % classification errors for reconstructed images of the simulated datasets as a function of

lesion size and contrast are presented in figure 6(a) for 64 mm3 and (b) for 8 mm3, for the FHMC and

the threshold based method (T42). A breakdown, in terms of PCEs and NCEs, of the % classification

errors in figure 6(a) is given in figures 7(a)-(c) for the FHMC, HMC and T42 segmentation methods

respectively. Finally, in order to facilitate a comparison of the segmentation results between the

simulated and the acquired datasets, the % computed volume error is given in figures 8(a)-(b) for the

same configurations as in figures 6(a)-(b).

Considering the simulated datasets, the introduction of FHMC led to superior results in

comparison to the current “gold standard” in functional volume delineation of T42. FHMC

segmentations led to <25% classification errors in computed volumes for lesions sizes >13mm

irrespective of contrast ratio, level of noise or lesion size. Errors of more than 200% for FHMC were

only observed for the 10mm sphere. Results for the T42 were more dependent on the lesion size,

relative to FHMC results, varying from 10% to more than 200% (even for spheres up to 22mm in

diameter for a contrast of 4:1 and 64mm3 voxel size). However, the use of T42 was found to work well

for lesion sizes of >17mm and a lesion to background ratio of 8:1 with % classification errors of 20-

30%. On the other hand, for a lesion to background ratio of 4:1, the T42 threshold led to over 100%

overestimation in the functional volume for lesions <28mm in diameter. As the errors bars in the

different figures reveal, there was a larger dependence on the statistical quality of the reconstructed

images observed with T42 in comparison to FHMC for the majority of the lesion sizes and contrast

configurations considered. In particular this was true for all of the lesions for a contrast ratio of 4:1

and for lesions <22mm for a contrast ratio of 8:1. For example, for the 17mm sphere and a contrast

15

ratio of 8:1, T42 resulted in classification errors of 20 to 35% whereas FHMC classification errors

from 15 to 17% were observed (fig. 6). On the other hand in the case of the 28mm sphere and a

contrast ratio of 4:1, T42 errors were ranging from 85 to 110% whereas FHMC resulted in errors of

17-18%. The reduction in the reconstruction voxel size (from 64 mm3 to 8 mm3) led to small

differences in the functional volumes determined using the FHMC segmentation algorithm, and

although it led to improvements in the T42 based segmented volumes, the % classification errors

remained at 80-200%. The trend observed with the standard voxel sizes on the variation of the

segmentation results as a function of statistical quality was similar for the reduced voxel size images.

For example in the case of the 22m sphere and a contrast ratio of 4:1 errors of 77-100% and 26-27%

were observed for T42 and FHMC respectively. In general, the largest errors were observed for the

smaller lesions of 10 and 13mm, where none of the segmentation algorithms considered performed

well under any of the configurations tested, with errors largely >200%. As shown in figure 7(a) FHMC

classification errors are essentially NCEs for the two biggest spheres and PCEs for the small ones. In

contrast, as shown in figures 7(b)-(c), T42 and HMC methods result essentially in PCEs, apart from

T42 in association with lesions >28mm in diameter and a lesion to background ratio of 8:1.

In terms of overall volume estimation errors on simulated datasets (see figures 8(a)-(b))

FHMC results on errors of up to 10% and between 10% and 20% for contrast ratio of 8:1 and 4:1

respectively, for lesions >13mm. T42 led to volume determination errors of <10% for lesions >17mm

in diameter and a lesion to background ratio of 8:1, while errors of over 100% where observed for

lesions <28mm with a lesion to background ratio of 4:1. However, while the lowest overall volume

error of T42 was around 10%, the corresponding classification error was >20%. In the case of an 8

mm3 reconstructed voxel size (figure 8(b)) small improvements were seen using the T42 for lesions ≥

13mm and >22mm for a lesion to background ratio of 8:1 and 4:1 respectively. Finally, no noticeable

differences were seen in the FHMC based segmentation results, apart from an improvement to <15%

in the volume estimation error for the 13mm lesion with a contrast size of 8:1.

Figures 9(a)-(b) show the results in terms of % error in the recovered activity as a function of

lesion size and contrast ratio considering the segmented volumes using 64 mm3 and 8 mm3

reconstructed voxel sizes. As it can be seen from this figure, FHMC and T50 led to the best results in

16

comparison to the “ground truth” throughout the different lesions sizes and contrasts evaluated,

although T50 introduces larger errors in comparison to the “ground truth” for lesion sizes of <22mm

and a contrast of 4:1. The use of the 8 mm3 voxels does not alter the conclusions as far as the

relationship between the results for the two methods evaluated is concerned, although in absolute

terms all algorithms perform worse in comparison to the results obtained for 64 mm3 voxels.

Considering the acquired datasets, figures 10(a)-(b) contains the results for the % overall

lesion volume estimation for the 64 mm3 and 8 mm3 voxels, while figures 11(a)-(b) show the

corresponding results for the activity quantitation errors. In terms of the volume estimation the general

trends were similar to those observed for the simulated datasets, with the FHMC performing better

than the T42 throughout the range of lesion sizes and contrasts evaluated. In absolute terms, the

FHMC results were better particularly in the case of 8 mm3 voxels where errors of <20% and 10%

were seen for lesions >10mm and >22mm respectively. T42 errors were similar to FHMC for the 8:1

ratio and spheres >13mm but ranged from 20 to >100% for the 4:1 ratio configuration. A larger

dependence to the statistical quality of the reconstructed images can be observed with the acquired

datasets, demonstrating the more robust performance of the FHMC algorithm in comparison to the

T42 methodology which was seen to be more affected by the images’ statistical quality. Using again

the example of the 22mm sphere (figure 10(a)), T42 errors were from 30 to 95% while FHMC errors

were less than 5%. Although the variation of the FHMC results was higher for smaller spheres (10 and

13 mm), it was still smaller than in the case of the T42 results. For example, FHMC applied to the

13mm sphere with a 4:1 contrast ratio (figure 10(b)) resulted in errors between 5 and 30% whereas

T42 errors ranged from 50 to 150%. Similar results between the FHMC and the T50 algorithms were

globally seen in terms of the % accuracy of the recovered activity concentration, confirming the trends

observed with the simulated datasets. Finally, similarly with the volume estimation, better results were

seen with the 8 mm3 reconstructed voxel’s size for both the T50 and the FHMC leading to activity

concentration estimation errors of between +10% and -10% for lesions >17mm in diameter.

4. Discussion

17

Although PET imaging applications are currently, in their majority, diagnostic and largely

based on visual interpretation, there is increasing interest in applications such as the use of PET for

radiotherapy treatment planning, as well as response to therapy and outcome prediction studies where

accurate functional volume and concentration of activity estimation respectively are indispensable.

Current state of the art methodologies for functional volume determination involve the use of adaptive

thresholding based on anatomical information or phantom studies. The performance of these

techniques is greatly dependent on lesion contrast and image noise characteristics and as this work has

demonstrated can lead to variable performance. On the other hand, already proposed automatic

segmentation methodologies have been mostly evaluated for use in lesion detection rather than lesion

volume determination. In addition, their performance is highly dependent, similarly to the thresholding

algorithms, on image contrast and noise characteristics.

Hidden Markov Chains is an automatic segmentation algorithm that allows noise modelling in

the images but has also previously been evaluated for lesion detection rather than functional volume

estimation. In the presented work a new algorithm (Fuzzy HMC) has been introduced and evaluated

allowing the incorporation within Hidden Markov Chains of a finite number of fuzzy levels in

combination with the “hard” classes considered in HMC, adding this way an estimation of imprecision

that can account for the effects of limited spatial resolution in emission tomography images. During

the evaluation of the FHMC, the inclusion of more than 2 fuzzy levels was found to not substantially

alter the segmentation results, while only the inclusion of the voxels belonging to the first fuzzy level

led to the most accurate results in terms of functional volume calculations throughout the range of

configurations considered. Although it would be possible to consider the use of HMC with four hard

classes and an additional rule to cluster the resulting segmentation map, the fuzzy nature of the borders

leads to computation issues in transition probabilities that HMC is not able to deal with. Note that the

significant addition of the fuzzy measure and mathematical changes in the model allows FHMC to

take into account such a configuration, mainly due to the fact that one given voxel can contain both

classes.

Finally, in this paper we have introduced the concept of classification errors rather than

volume estimation errors in the evaluation of segmentation algorithms for volume determination tasks.

18

An evaluation based on classification errors is more robust since it does not simply consider the

segmented volume but also its location with respect to the “ground truth” known in simulated datasets.

Therefore, while the absolute segmented volume may be correct its location may be wrong, a fact that

is as significant as the correct estimation of the overall functional volume particularly for applications

such as the use of PET volumes in radiotherapy treatment planning.

In comparison to the recommended T42 for the accurate functional volume determination in

PET (Krak et al 2005), the FHMC algorithm gave clearly superior results for lesions <28mm, in

particular considering a lesion contrast of 4:1, where the T42 methodology completely failed to

recover the functional volume. In addition, FHMC was more robust considering the different image

statistical quality levels evaluated, while the results of T42 were greatly influenced by the level of

noise present in the images. Differences between classification and volume estimation errors across

the different configurations evaluated were smaller for the segmented volumes provided by the FHMC

algorithm. In addition, the classification error results allows us to establish that the accuracy obtained

on the estimated volumes using the FHMC algorithm is not by chance due to a similar levels of

negative and positive classification errors. A smaller reconstructed voxel size at the same statistical

quality level led to worse overall segmentation results, without altering the conclusions as far as the

relative performance of the different algorithms evaluated is concerned.

The performance of the segmentation algorithms under evaluation in the reconstructed images

of the acquired datasets was similar to that obtained with the simulated datasets in terms of volume

estimation errors. The only difference observed in comparison with the simulated dataset results was

an improvement in the overall segmentation results for 8 mm3 reconstructed images in comparison to

the 64 mm3, which can be attributed to an associated adjustment of the optimised reconstruction

parameters as a function of the voxel size in the 3D RAMLA algorithm used to reconstruct the

acquired datasets.

As far as concentration recovery results are concerned, the performance of the different

segmentation algorithms was compared in the simulated datasets to the recovered activity

concentration considering the exact size and location of the simulated lesions. These results were

influenced by the effects of partial volume leading to increasing underestimation of the activity

19

concentration with decreasing lesion size. Segmentation algorithms concentrate on accurate edge

modelling in the object of interest and do not as such account for changes in the values of the voxels as

a result of PVE. FHMC and the current “state of the art” threshold of 50% of the maximum lesion

value (Krak et al 2005) led to similar results independently of the configurations evaluated, with

absolute differences of 10-15% (due to an extra underestimation for the T50 results). Similar trends to

those observed with the simulated datasets were obtained from the segmentation of the acquired

images.

The presented results demonstrate the interest of FHMC over thresholding algorithms as the

flexibility of the fuzzy levels choice may allow the use of the same segmentation map for different

tasks, across a large range of lesion contrasts and sizes. FHMC through the addition of the fuzzy levels

associated with each hard class is able to more accurately model the object of interest edges in

reconstructed PET images. In addition, FHMC is clearly less susceptible to alterations in statistical

image quality and lesion contrasts than other methodologies. This was observed on both images of

simulated and acquired datasets. Having said that, none of the evaluated algorithms was successful in

accurate volume estimation for lesion sizes of <17mm, considering typical PET image statistical

qualities and reconstructed voxels of either 8mm3 or 64mm3. The main reason behind the failure of

FHMC concerning the segmentation of such small lesions is the small number of voxels associated

with the object of interest in combination to image noise levels, and the Hilbert-Peano path used to

transform the image into a chain. The spatial correlation of such small objects may be lost once the

image is transformed into a chain. A local model may be able to overcome such an issue (Hatt M et al

2007).

The results for FHMC may be further improved. Firstly, the direct estimation of the noise in

the reconstructed images may lead to better results in comparison to the assumed Gaussian model used

in this work to fit the distribution for each of the classes. Secondly, other a priori models may be used

for Markovian modelling, like couple (Pieczynski et al 2004) or triplet (Lanchantin et al 2004) Markov

Chains or Fields. These may be of interest considering a better modelling of the transitions between

boundaries classes, as well as the non stationary nature of the hidden a priori model. In addition, the

20

fuzzy model may be extended to more than two hard classes to better model inhomogeneous or non

spherical objects of interest.

5. Conclusion

A modified version of Hard Markov Chains segmentation algorithm has been developed by

introducing a fuzzy measure (FHMC). Our results with both simulated and acquired datasets have

shown that FHMC is more effective than these of the reference thresholding methodologies for both

VOI determination and quantification in PET imaging. As part of the evaluation process, we have also

introduced and assessed the interest of classification errors as a robust measurement of the

performance of segmentation algorithms for VOI determination in contrast to a simple volume

estimation which may introduce biases in terms of the segmented lesion location. Future developments

will concentrate on the use of more than two “hard” classes in FHMC, which may more accurately

account for the presence of inhomogeneous or non-spherical functional volumes, as well as an

investigation into more adequate noise and a priori models.

21

APPENDIX. The FHMC algorithm step by step

For the calculation of the expressions a quantization of the interval [ ]0,1 into intervals

1 2 10, , ..., ,1

N

N N N

−

is used. For example with 2 fuzzy levels (or intervals) 1F , 2F ,we have 3N = and

there are 1 2N − = fuzzy levels with i

i

Nε = : 1

1

3ε = and 2

2

3ε = . Note : the symbol .... denotes a

density instead of a probability. 1. Transformation of the 2D or 3D image in a 1D chain using the Hilbert-Peano path (Kamata et al 1999) (save the path to be used in step 5 of the procedure).

From this point on, every step is performed on the image transformed into a chain.

2. Parameters initialization:

A priori model parameters:

1 2

1 2

0.25( ) and ( , )

0init c trans c d

α α

γ γ

= =

= = are computed according to (3), (4) and the following:

1 1 1 1

21 1 1 1

2 11 1 | 1| 1 | |

N N N N

i i i j

i i i j

N N N N N Nλ

− − − −

= = = =

= − + − − + − −

∑ ∑ ∑∑ 1 2 1 21 ( )α α γ γ

βλ

− + + +=

Initial and transition probabilities initializations can then be computed as follows:

( ) ( )

1

1 11

1

2 21

1

1

(0) (1 )

(1) (1 |1 |)

1( ) 1 1 | 1| 1 | |

N

i

N

i

N

i i i i

j

iinit

N N

iinit

N N

jinit

N N N

βα γ

βα γ

βε ε ε ε

−

=

−

=

−

=

= + + −

= + + − −

= − + − − + − −

∑

∑

∑

] [

1 if 0,1( , )

( , ) with 1( ) if 0,1

C dg c d

trans c d Ch d C d

N

= ∈

= = ∈

Noise model parameters:

0 1 0 1( , , , ) ( , 2)KMeans Yµ µ σ σ = with Y the image and 2 for the two hard classes to look for.

Then we determine parameters of each fuzzy level with (9). 3. SEM procedure for parameters estimation

At each iteration q until no significant modification of the estimated parameters (convergence):

a. fwd and bwd densities computation for each class c , 0,1, , 1,..., 1ic i Nε∈ = − is performed

recursively as follows:

For 1t = : 11( ) ( ) ( )cfwd c h c G y=

22

For 1t > : 1

1 1 11

1( ) ( ) (0) ( ,0) (1) ( ,1) ( ) ( , )

N

c t i it t t t

i

fwd c G y fwd trans c fwd trans c fwd trans cN

ε ε−

− − −=

= + +

∑

For t T= : ( ) 1Tbwd c =

For t T< : 10 1( ) ( ) (0) (0, )t ttbwd c G y bwd trans c++=

1

1 11 1 11

1( ) (1) (1, ) ( ) ( , ) ( )

i

N

t tt t i i

i

G y bwd trans c G y trans c bwdN

ε ε ε−

+ ++ +=

+ + ∑

These computations must be normalized. cG is given by:2

2

( )1( ) exp

22c

c

cc

yG y

µ

σσ π

−= −

b. Stochastic re-estimation of parameters:

To obtain one a posteriori realization of X , simulate a fuzzy Markov chain using the following : Posterior distributions of X are defined by:

1( ) ( )init c fwdbwd c= and

11 11

11

0

( | ) ( ) ( )( , )

( | ) ( ) ( ) ( )

tt d t

td t

f d c G y bwd dtrans c d

f d c G y bwd d dv d

++ +

++

=

∫

] [

(c) if 0,1( ) 1

(c) if 0,1

init c

init cinit c

N

∈

= ∈

and

] [

t(c,d) if 0,1( , ) 1

(c,d) if 0,1

t

trans d

trans c dtrans d

N

∈

= ∈

It has to be noted that ( , )ttrans c d depends on t since a different transition matrix is computed for each element of the posterior realization, as we are dealing with a non stationary Markov chain. The estimated values of the parameters at the iteration q are computed on the simulated a posteriori

chain | 1...tx t T= as follows:

For the a priori model:

[ ] [ ]

[ ]

[ ] [ ]

[ ]

1

12

12

( ) ( , )

( , ) ( , )( , )

( , )

q q

Tq q

t tq t

Tq

t

t

init c x c

x c x d

trans c d

x c

δ

δ δ

δ

−=

−=

=

=∑

∑

For the noise model: [ ]

[ ]

[ ]

1

1

( , )

( , )

Tq

t tq t

c Tq

t

t

y x c

x c

δ

µ

δ

=

=

=∑

∑ [ ]

[ ] [ ]

[ ]

2

2 1

1

( , )( )

( , )

Tq q

t t cq t

c Tq

t

t

x c y

x c

δ µ

σ

δ

=

=

−

=∑

∑

for 0c = and 1c = . For fuzzy levels ( ic ε= ) noise parameters, use

equation (9)

with 1 if

( , )0 if

m nm n

m nδ

==

≠

4. MPM segmentation of the chain using estimated parameters:

For each tx , determine the class (hard class or fuzzy level) minimizing the error classification

probability by minimizing the following expression:

23

ˆ ˆ(0) (0, ( )) (1) (1, ( ))t tt tfwdbwd L s y fwdbwd L s y+

1

0

ˆ( ) ( , ( ))i i t it

fwdbwd L s y dε ε ε+∫

for every ( )ts y , and where fwdbwd denotes the product of the forward and backward densities. The

cost function L is given by (8). 5. Reverse transformation of the 1D segmented chain into the 2D or 3D segmentation map using the path saved at step 1.

24

Acknowledgments

This work is financially supported by a Region of Brittany research grant under the “Renouvelement

des competences” program 1202-2004.

25

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28

Figure Captions

Figure 1: The 3D Hilbert-Peano space filling curve for a 4× 4× 4 voxels VOI. Figure 2: (a) A graphical representation of the IEC phantom, and the central slice of the digital IEC phantom used in the generation of the simulated datasets (b) with 2× 2× 2 mm3 and (c) 4× 4× 4 mm3. Figure 3: Different images used in the segmentation study; (a)-(d) simulated : (a) ratio 4:1, 20 millions of coincidences, 64 mm3, (b) ratio 8:1, 40 millions, 64 mm3, (c) ratio 4:1, 20 millions of coincidences, 8 mm3, (d) ratio 8:1, 40 millions, 8 mm3 (e)-(h) acquired : (e) ratio 4:1, 2min acquisition time, 64 mm3, (f) ratio 8:1, 5min, 64 mm3, (g) ratio 4:1, 5min, 8 mm3, (h), ratio 8:1, 5min, 8 mm3. Figure 4: Examples of segmentation maps for the 28mm sphere (one slice) (a) PET ROI, (b) Digital “ground truth”, (c) HMC map, (d) T42 map, (e) FHMC with 2 fuzzy levels (light and dark grey voxels) segmentation map, (f) Map used for VOI (hard class + 1st fuzzy level, FHMC 1/2), (g) Map for quantitation (only hard class voxels, FHMC 0/2), (h) T50 map. Note that in this particular case, FHMC 1/2 for VOI and T50 result in the same map but this is of course not always the case (especially having considered the complete 3D volume). Figure 5: (a) Classification errors for the lesion VOI determination and (b) lesion activity recovery errors in the simulated images for the FHMC vs HMC segmentation. Different number of fuzzy levels (2 or 3) were used in the segmentation process and different number of these were subsequently selected to (a) form the segmented volumes or (b) determine lesion average activity concentrations for the different imaging conditions considered. Figure 6: Classification errors in lesion VOI determination from the simulated images: (a) 64 mm3 voxels and (b) 8 mm3 voxels for the FHMC 1/2 vs T42 segmentation under variable imaging conditions. Figure 7: Repartition of PCEs and NCEs from the (a) FHMC 1/2, (b) HMC and (c) T42 segmentation results for the different simulated imaging configurations considered. Figure 8: Lesion VOI estimation errors from the simulated images: (a) 64 mm3 voxels and (b) 8 mm3 voxels for the FHMC 1/2 vs T42 segmentation under variable imaging conditions. Figure 9: Lesion average activity concentration estimation errors from the simulated images: (a) 64 mm3 voxels and (b) 8 mm3 voxels for FHMC 0/2 vs T50 segmentation under variable imaging conditions. Figure 10: Lesion VOI estimation errors from the acquired images: (a) 64 mm3 voxels and (b) 8 mm3 voxels for the FHMC 1/2 vs T42 segmentation under variable imaging conditions. Figure 11: Lesion average activity concentration estimation errors from the acquired images: (a) 64 mm3 voxels and (b) 8 mm3 voxels for the FHMC 0/2 vs T50 segmentation under variable imaging conditions.

29

Figure 1

30

(a)

(b) (c)

Figure 2

31

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Figure 3

32

(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 4

33

0

20

40

60

80

100

120

140

Configurations

Cla

ss

ific

ati

on

err

or

(%)

FHMC 1 / 2 fuzzy levelsFHMC 1 / 3 fuzzy levelsFHMC 2 / 3 fuzzy levelsFHMC 2 / 2 fuzzy levelsFHMC 3 / 3 fuzzy levelsHMC

37mm28mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm28mm17mm10mm 13mm 22mm

(a)

-80

-70

-60

-50

-40

-30

-20

-10

0

Configurations

Qu

an

tita

tio

n e

rro

rs (

%)

Ground truth FHMC 0 / 2 FHMC 1 / 2

FHMC 2 / 2 HMC

37mm28mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm28mm17mm10mm 13mm 22mm

(b)

Figure 5

34

0

20

40

60

80

100

120

140

Configurations

Cla

ssif

ica

tio

n E

rro

r (%

)

FHMC T42

37mm28mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm28mm17mm10mm 13mm 22mm

(a)

0

20

40

60

80

100

120

140

Configurations

Cla

ss

ific

ati

on

Err

or

(%)

FHMC T42

37mm28mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm28mm17mm10mm 13mm 22mm

(b)

Figure 6

35

PCEs and NCEs

of FHMC (1 / 2) segmentation method

0

20

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Configurations

Err

or

(%),

wit

h r

esp

ect

to

nu

mb

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of

vo

xels

in

th

e s

ph

ere

PCEs NCEs

37mm28mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm28mm17mm10mm 13mm 22mm

Cla

ssific

atio

n e

rro

rs (

%)

(a)

PCEs and NCEs

of HMC segmentation method

0

20

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Configurations

Err

or

(%),

wit

h r

esp

ect

to

nu

mb

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of

vo

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in

th

e s

ph

ere

PCEs NCEs

37mm28mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm28mm17mm10mm 13mm 22mm

Cla

ssific

ation e

rrors

(%

)

(b)

36

PCEs and NCEs

of T42 segmentation method

0

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Configurations

Err

or

(%),

wit

h r

esp

ect

to

nu

mb

er

of

vo

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in

th

e s

ph

ere

PCEs NCEs

37mm28mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm28mm17mm10mm 13mm 22mm

Cla

ssific

atio

n e

rro

rs (

%)

(c)

Figure 7

37

0

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140

Configurations

Vo

lum

e E

rro

r (%

)

FHMC T42

37mm28mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm28mm17mm10mm 13mm 22mm

(a)

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r (%

)

FHMC T42

37mm28mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm28mm17mm10mm 13mm 22mm

(b)

Figure 8

38

-90

-70

-50

-30

-10

Configurations

Qu

an

tita

tio

n e

rro

r (%

)Ground truth FHMC T50

37mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm17mm10mm 13mm 22mm28mm 28mm

(a)

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

Configurations

Qu

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tita

tio

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rro

r (%

)

Ground truth FHMC T50

37mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm17mm10mm 13mm 22mm28mm 28mm

(b)

Figure 9

39

0

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Configurations

Vo

lum

e E

rro

r (%

)

FHMC T42

37mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm17mm10mm 13mm 22mm

(a)

0

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60

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140

Configurations

Vo

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rro

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)

FHMC T42

37mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm17mm10mm 13mm 22mm

(b)

Figure 10

40

-80

-70

-60

-50

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-30

-20

-10

Configurations

Qu

an

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tio

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rro

r (%

)

FHMC T50

37mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm17mm10mm 13mm 22mm

(a)

-70

-60

-50

-40

-30

-20

-10

0

10

Configurations

Qu

an

tita

tio

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rro

r (%

)

FHMC T50

37mm17mm10mm

Ratio 4:1 Ratio 8:1

13mm 22mm 37mm17mm10mm 13mm 22mm

(b)

Figure 11

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